Transcript Chapters 5-7 - ShareStudies.com

Chapter 5
Review
Based on your assessment of the stock
market, you state that the chances that
stock prices will start to go down within
2 months are 50-50. This concept of
probability based on your beliefs is
called
a.
b.
c.
d.
Classical probability
Empirical probability
Subjective probability
Independence
Based on your assessment of the stock
market, you state that the chances that
stock prices will start to go down within
2 months are 50-50. This concept of
probability based on your beliefs is
called
a.
b.
c.
d.
Classical probability
Empirical probability
Subjective probability
Independence
A study of absenteeism from the
classroom is being conducted. In terms
of statistics, the study is called
a.
b.
c.
d.
An experiment
An event
An outcome
A joint probability
A study of absenteeism from the
classroom is being conducted. In terms
of statistics, the study is called
a.
b.
c.
d.
An experiment
An event
An outcome
A joint probability
In a study of absenteeism, the results
showed that 126 students were absent
from Monday morning classes. This
number (126) is called
a.
b.
c.
d.
An experiment
An event
An outcome
A joint probability
In a study of absenteeism, the results
showed that 126 students were absent
from Monday morning classes. This
number (126) is called
a.
b.
c.
d.
An experiment
An event
An outcome
A joint probability
To apply this rule of addition,
P(A
or B)= P(A) + P(B), the events must be
a.
b.
c.
d.
Joint events
Conditional events
Mutually exclusive events
Independent events
To apply this rule of addition,
P(A or B)= P(A) + P(B), the events must
be
a.
b.
c.
d.
Joint events
Conditional events
Mutually exclusive events
Independent events
Management claims that the probability
of a defective relay is only 0.001. The
rule used for finding the probability of
the relay not being defective is the
a.
b.
c.
d.
Addition rule
Multiplication rule
Complement rule
Special rule of addition
Management claims that the probability
of a defective relay is only 0.001. The
rule used for finding the probability of
the relay not being defective is the
a.
b.
c.
d.
Addition rule
Multiplication rule
Complement rule
Special rule of addition
Management claims that the probability
of a defective relay is only 0.001. The
probability of the relay not being
defective is
a.
b.
c.
d.
0.002
0.000001
0.999
1.0
Management claims that the probability
of a defective relay is only 0.001. The
probability of the relay not being
defective is
a.
b.
c.
d.
0.002
0.000001
0.999
1.0
Probability Problem Solving Process

For each problem



Determine what the events are.
Decide whether to use the “and”
combination, the “or” combination, or the
complement rule.
Before choosing the correct formula, ask
one of the following questions:
1. For the AND combination: are the events
independent?
2. For the OR combination: are the events
mutually exclusive?

Write down the appropriate formula.
a.
b.
c.
d.
If you roll a single die and count the number of
dots on top, what is the sample space of all
possible outcomes? Are the outcomes equally
likely?
Assign probabilities to the outcomes of the
sample space in part a. Do the probabilities
add up to 1?
What is the probability of getting a number
less than 5 on a single throw?
What is the probability if getting 5 or 6 on a
single throw?
a.
The sample space consists of equally likely
outcomes of {1,2,3,4,5,6}
b.
Assign each of the 6 outcomes a probability of
1/6. Probabilities add up to:
1/6+1/6+1/6+1/6+1/6+1/6=6/6=1
c.
Being successful in getting a number less than
5 on a single throw can be achieved by getting
a 1 or a 2 or a 3 or a 4. Since there are 4 ways
of being successful out of 6 possible ways, the
probability is 4/6 (or 2/3).
d.
Since there are 2 ways (rolling a 5 or a 6) of
being successful out of the 6 possible ways, the
probability of getting a 5 or a 6 on a single
throw is 2/6 or 1/3.
Two marbles are drawn without
replacement from a bowl containing 7
red, 4 blue, and 9 yellow marbles.
1.
2.
3.
What is the probability that both are
red?
What is the probability neither is
red?
What is the probability at least one is
red?
1. (both red)
a) Events are: A= first marble is red
B= second is red
b) This is an “and” combination (1st & 2nd are red).
c) To determine the correct formula, we need to
know “are the events independent or
dependent?”
d) The events are dependent since the 1st marble
is not replaced before the 2nd is drawn.
e) Use the formula P(A and B)= P(A)P(B|A)
P(1st is red)=P(A)= 7/20;
P(2nd is red, given 1st is red)= P(B|A)= 6/19
P(A and B)= (7/20)(6/19)= .1105
2. (neither red)
a) This is an AND combination: the 1st is not red
and the 2nd is not red.
b) The events are dependent since the 1st marble
is not replaced before the drawing the 2nd .
c) P(1st not red)= P(A)= 13/20;
P(2nd not red, given 1st not red)= P(B|A)= 12/19
d) Using the multiplication rule:
P(neither is red)= (13/20)(12/19)= .4105
3. (at least one red)
a) The complement to “at least one is red” is
“neither”.
b) Using the complement rule:
P(at least 1 is red)= 1-P(neither is red)
= 1- .4105
= .5895
2 dice are rolled and the sum of the
dice is noted.
4.
5.
6.
What is the probability the sum is a 7
or 11?
What is the probability it is at least a
ten?
What is the probability the sum is
less than a ten?
4. (sum 7 or 11)
a)
This is an OR combination.
b)
The events are mutually exclusive,
since a roll cannot be both a 7 and
an 11 at the same time.
c)
Use the simple addition rule:
P(A or B)= P(A) + P(B)
P(7 or 11)= (6/36) + (2/36)= .2222
5. (sum at least 10)
a)
This is an OR combination. At least a 10
means 10 OR 11 OR 12. These 3
outcomes are mutually exclusive, so we
may add the probabilities.
b)
P(at least a 10)= 3/36 + 2/36 + 1/36
= 6/36
= .1667
6.
a) This is the complement to number 5
above.
b) P(less than 10)= 1- P(at least 10)
= 1- .1667
= .8333
Respondents in 3 cities were asked whether they
would buy a new breakfast cereal that was being taste
tested. Use the responses from the contingency table
to answer the questions.
Yes
No
Undecided
Total
Fort
Worth
100
Dallas
Chicago
Total
150
150
400
125
130
95
350
75
170
5
250
300
450
250
1000
7.
What is the probability a person
responded yes?
Total in Yes row is 400 and in sample is
1000.
Probability a person responded yes is
400/1000= .4
8.
What is the probability a person is
from Dallas?
Total in Dallas row is 450 and in sample
is 1000.
Probability a person is from Dallas is
450/1000= .45
9.
What is the probability a person
responded yes and is from Dallas?
Categories intersect. The number in the
intersecting cell is 150.
Probability a person responded yes and
is from Dallas is 150/1000= .15
10.
What is the probability a person
responded yes or is from Dallas?
Categories are not mutually exclusive.
There are 400 who responded yes and
450 from Dallas, however 150 of
them are in both categories.
Probability a person responded yes or is
from Dallas is (.4)+ (.45)- (.15)= .7
11.
What is the probability a person
responded yes given they are from
Dallas?
Only refer to the Dallas column.
Probability a person responded yes
given they are from Dallas is
150/450= .33
12.
What is the probability a person is
from Dallas given they responded
yes?
Only refer to the Yes row.
Probability a person is from Dallas given
they responded yes is 150/400= .375
In the book Chances: Risk and Odds of every day life,
James Burke says that 56% of the general population
wears eyeglasses, while only 3.6% wears contacts. He
also noted that of those who do wear glasses, 55.4%
are women and 44.6% are men. Of those who wear
contacts, 63.1% are women and 36.9% are men.
Assume that no one wears both glasses and contacts.
For the next person you encounter at random, what is
the probability that this person is
13.
14.
15.
16.
17.
A woman wearing glasses?
A man wearing glasses?
A woman wearing contacts?
A man wearing contacts?
None of the above?
First symbolize the given information:
P(person wears glasses)= 56%
P(woman, given wears glasses)= 55.4%
P(man, given wears glasses)= 44.6%
P(person wears contacts)= 3.6%
P(woman, given wears contacts)= 63.1%
P(man, given wears contacts)= 36.9%
A woman wearing glasses: P(wears glasses and woman)
use multiplication rule:
P(person wears glasses)P(woman, given wears glasses)=
(56%)(55.4%)=.31 or 31%
13.
First symbolize the given information:
P(person wears glasses)= 56%
P(woman, given wears glasses)= 55.4%
P(man, given wears glasses)= 44.6%
P(person wears contacts)= 3.6%
P(woman, given wears contacts)= 63.1%
P(man, given wears contacts)= 36.9%
A man wearing glasses: P(wears glasses and man)
use multiplication rule:
P(person wears glasses)P(man, given wears glasses)=
(56%)(44.6%)=.25 or 25%
14.
First symbolize the given information:
P(person wears glasses)= 56%
P(woman, given wears glasses)= 55.4%
P(man, given wears glasses)= 44.6%
P(person wears contacts)= 3.6%
P(woman, given wears contacts)= 63.1%
P(man, given wears contacts)= 36.9%
A woman wearing contacts: P(wears contacts and
woman)
use multiplication rule:
P(person wears contacts)P(woman, given wears contacts)=
(3.6%)(63.1%)=.023 or 2.3%
15.
First symbolize the given information:
P(person wears glasses)= 56%
P(woman, given wears glasses)= 55.4%
P(man, given wears glasses)= 44.6%
P(person wears contacts)= 3.6%
P(woman, given wears contacts)= 63.1%
P(man, given wears contacts)= 36.9%
A man wearing contacts: P(wears contacts and man)
use multiplication rule:
P(person wears contacts)P(man, given wears contacts)=
(3.6%)(36.9%)=.013 or 1.3%
16.
First symbolize the given information:
P(person wears glasses)= 56%
P(woman, given wears glasses)= 55.4%
P(man, given wears glasses)= 44.6%
P(person wears contacts)= 3.6%
P(woman, given wears contacts)= 63.1%
P(man, given wears contacts)= 36.9%
Neither.
First compute P(person wears contacts OR glasses) by
adding the probabilities 56% and 3.6% (=59.6%). 59.6%
of the population wears either glasses or contacts. Using
the complement rule we can find the probability that the
person does not wear either:
100% - 59.6%= 40.4% (or in decimal form:1-.596=.404)
17.
Wing Foot is a shoe franchise commonly found in shopping
centers across the U.S. Wing Foot knows that its stores will
not show a profit unless they make over $540,000 per year.
Let A be the event that a new Wing Foot store makes over
$540,000 its first year. Let B be the event that a store makes
more than $540,000 its second year. Wing Foot has an
administrative policy of closing a new store if it does not show
a profit in either of the first two years. The accounting office at
Wing Foot provided the following information: 65% of all Wing
Foot stores show a profit the first year; 71% of all Wing Foot
stores show a profit the second year (this included stores that
did not show a profit in the first year); however, 87% of Wing
Foot stores that showed a profit the first year also showed a
profit the second year. Compute the following:
a) P(A); b) P(B); c) P(B, given A); d) P(A and B); e) P(A or B)
f) What is the probability that a new Wing Foot store will not be
closed after 2 years? What is the probability that a new Wing
Foot store will be closed after 2 years?
a)
b)
c)
d)
P(A)= .65
P(B)= .71
Since the problem states that 87% of Wing
Foot stores that showed a profit the first
year also showed a profit the second year:
P(B, given A)= .87
The events A and B are dependent
[P(B)≠P(B|A)].
So P(A and B)= P(A)P(B|A)
=(.65)(.87)
=.5655
e)
f)
The events A and B are not mutually
exclusive. There were stores successful in
both years.
So, P(A or B)= P(A)+ P(B)- P(A and B)
= .65 + .71 - .5665
= .7945
The administrative policy is to close a
store if it does not show a profit in either of
the first two years. Therefore, there is a
probability of 79.45% that a new Wing
Foot store will not be closed after 2 years.
There is a probability of (1-.7954=.2055)
20.55% that a new store will be closed
after two years.